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Central Configurations for Newtonian -Body Problems
We show the existence of spatial central configurations for the -body problems. In the -body problems, N bodies are at the vertices of a regular N-gon T ; 2p bodies are symmetric with respect to the center of T, and located on the straight line which is perpendicular to the regular N-gon T and passes through the center of T; the th is located at the the center of T. The masses located on the vertices of the regular N-gon are assumed to be equal; the masses located on the same line and symmetric with respect to the center of T are equal.
1. Introduction and Main Results
The Newtonian -body problems [1–3] concern with the motions of particles with masses and positions , and the motion is governed by Newton’s second law and the Universal law: where and with Newtonian potential: Consider the space that is, suppose that the center of mass is fixed at the origin of the space. Because the potential is singular when two particles have the same position, it is natural to assume that the configuration avoids the collision set for some . The set is called the configuration space.
Definition 1 (see [2, 3]). A configuration = is called a central configuration if there exists a constant such that The value of constant in (4) is uniquely determined by where Since the general solution of the -body problem cannot be given, great importance has been attached to search for particular solutions from the very beginning. A homographic solution is a configuration which is preserved for all time. Central configurations and homographic solutions are linked by the Laplace theorem . Collapse orbits and parabolic orbits have relations with the central configurations [2, 4–6], so finding central configurations becomes very important. The main general open problem for the central configurations is due to Wintner  and Smale : is the number of central configurations finite for any choice of positive masses ? Hampton and Moeckel  have proved this conjecture for any given four positive masses.
For 5-body problems, Hampton  provided a new family of planar central configurations, called stacked central configurations. A stacked central configuration is one that has some proper subset of three or more points forming a central configuration. Ouyang et al.  studied pyramidal central configurations for Newtonian -body problems; Zhang and Zhou  considered double pyramidal central configurations for Newtonian -body problems; Mello and Fernandes  analyzed new classes of spatial central configurations for the -body problem. Llibre and Mello studied triple and quadruple nested central configurations for the planar -body problem. There are many papers studying central configuration problems such as [13–22].
Based on the above works, we study stacked central configuration for Newtonian -body problems. In the -body problems, bodies are at the vertices of a regular -gon , and bodies are symmetrically located on the same straight line which is perpendicular to and passes through the center of ; the th body is located at the center of . The masses located on the vertices of the regular -gon are equal; the masses located on the line and symmetric with respect to the center of are equal. (see Figure 1 for and ).
In this paper we will prove the following result.
Theorem 2. For -body problem in where and , there is at least one central configuration such that bodies are at the vertices of a regular -gon , and 2 bodies are symmetric with respect to the center of the regular -gon , and located on a line which is perpendicular to the regular -gon ; the th body is located at the center of . The masses at the vertices of are equal and the masses symmetric with respect to the center of are equal.
2. The Proof of Theorem 2
2.1. Equations for the Central Configurations of -Body Problems
To begin, we take a coordinate system which simplifies the analysis. The particles have positions given by , where , ; , , where ; .
The masses are given by , , where .
Notice that is a central configuration if and only if By the symmetries of the system, (7) is equivalent to the following equations: that is, where In order to simplify the equations, we defined , , where ; , when ; , when ; when ; , , where .Equations (9)–(11) can be written as a linear system of the form given by The column vector is given by the variables . Since the is function of , we write the coefficient matrix as .
We need the next lemma.
Lemma 3 (see ). Assuming , , there is a nonempty interval , and , such that for each , forms a central configuration of the -body problem.
2.3. For All
The proof for is done by induction. We claim that there exists such that system (12) has a unique solution , for . We have seen that the claim is true for . We assume the claim is true for and we will prove it for . Assume by induction hypothesis that there exists such that system (12) has a unique solution and for .
We need the next lemma.
Lemma 4. There exists such that , for and is a solution of (12).
Proof. Since , we have that the first equation of (12) is satisfied when , for and . Substituting this solution into the last equation of (12), we let We have that Therefore there exists at least a value satisfying equation . This completes the proof of Lemma 4.
Let , we define It is not difficult to see that the system (12) is equivalent to for .
Let be the solution of system (12) given in Lemma 4. The differential of (17) with respect to the variables is We have assumed that exists such that system (12) with instead of has a unique solution, so ; therefore . Applying the Implicit Function Theorem, there exists a neighborhood of , and unique analytic functions , for and , such that is the solution of the system (12) for all . The determinant is calculated as where is the algebraic cofactor of .
We see that , and for do not contain the factor . If we consider as a function of , then is analytic and nonconstant. We can find sufficiently close to such that and therefore, a solution of system (12) is satisfying , for .
The proof of Theorem 2 is completed.
The authors express their gratitude to Professor Zhang Shiqing for his discussions and helpful suggestions. This work is supported by NSF of China and Youth found of Mianyang Normal University.
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Copyright © 2013 Furong Zhao and Jian Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.